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Calculus · Definite-Integrals

Integrate x^2·e^(-x^2/4)/(2·sqrt(pi)) from -oo to oo

\int_{-\infty}^{\infty} \frac{x^{2} e^{- \frac{x^{2}}{4}}}{2 \sqrt{\pi}}\, dx

Step by step

  1. \int_{-\infty}^{\infty} \frac{x^{2} e^{- \frac{x^{2}}{4}}}{2 \sqrt{\pi}}\, dx

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. \int \frac{x^{2} e^{- \frac{x^{2}}{4}}}{2 \sqrt{\pi}}\, dx = \frac{- 2 x e^{- \frac{x^{2}}{4}} + 2 \sqrt{\pi} \operatorname{erf}{\left(\frac{x}{2} \right)}}{2 \sqrt{\pi}}

    This one needs a special technique; the computer algebra system gives the antiderivative directly.

  3. F(\infty) - F(-\infty) = \left(\text{NaN}\right) - \left(\text{NaN}\right)

    Fundamental theorem of calculus: plug in the limits.

  4. = 2

    Simplify.

Reveal the answer
2